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A priori error analysis for HDG methods using extensions from subdomains to achieve boundary conformity. (English) Zbl 1290.65110

The a priori error analysis of a technique for numerical solution of steady-state diffusion problems on curved domains is presented. It is shown that the order of convergence in the \(L^2\)-norm of the approximate flux and scalar unknowns is optimal if the distance between the boundary of the original domain and the computational domain is of order \(h\). Further, the superconvergence of the projection of the error of the scalar variable is proven. The numerical experiments confirming the theoretical results are shown.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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[1] D. Arnold, F. Brezzi, B. Cockburn and D. Marini, Unified analysis of discontinuous Galerkin methods for second-order elliptic problems, SIAM J. Numer. Anal. 39 (2002), 1749-1779. · Zbl 1008.65080
[2] John W. Barrett and Charles M. Elliott, Finite element approximation of the Dirichlet problem using the boundary penalty method, Numer. Math. 49 (1986), no. 4, 343 – 366. · Zbl 0614.65116
[3] John W. Barrett and Charles M. Elliott, Fitted and unfitted finite-element methods for elliptic equations with smooth interfaces, IMA J. Numer. Anal. 7 (1987), no. 3, 283 – 300. · Zbl 0629.65118
[4] J. Thomas Beale and Anita T. Layton, On the accuracy of finite difference methods for elliptic problems with interfaces, Commun. Appl. Math. Comput. Sci. 1 (2006), 91 – 119. · Zbl 1153.35319
[5] R. P. Beyer and R. J. LeVeque, Analysis of a one-dimensional model for the immersed boundary method, SIAM J. Numer. Anal. 29 (1992), no. 2, 332 – 364. · Zbl 0762.65052
[6] James H. Bramble and J. Thomas King, A robust finite element method for nonhomogeneous Dirichlet problems in domains with curved boundaries, Math. Comp. 63 (1994), no. 207, 1 – 17. · Zbl 0810.65104
[7] James H. Bramble and J. Thomas King, A finite element method for interface problems in domains with smooth boundaries and interfaces, Adv. Comput. Math. 6 (1996), no. 2, 109 – 138 (1997). · Zbl 0868.65081
[8] Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. · Zbl 0788.73002
[9] Bernardo Cockburn and Bo Dong, An analysis of the minimal dissipation local discontinuous Galerkin method for convection-diffusion problems, J. Sci. Comput. 32 (2007), no. 2, 233 – 262. · Zbl 1143.76031
[10] Bernardo Cockburn, Jayadeep Gopalakrishnan, and Raytcho Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009), no. 2, 1319 – 1365. · Zbl 1205.65312
[11] Bernardo Cockburn, Jayadeep Gopalakrishnan, and Francisco-Javier Sayas, A projection-based error analysis of HDG methods, Math. Comp. 79 (2010), no. 271, 1351 – 1367. · Zbl 1197.65173
[12] Bernardo Cockburn, Deepa Gupta, and Fernando Reitich, Boundary-conforming discontinuous Galerkin methods via extensions from subdomains, J. Sci. Comput. 42 (2010), no. 1, 144 – 184. · Zbl 1203.65250
[13] Bernardo Cockburn, Johnny Guzmán, and Haiying Wang, Superconvergent discontinuous Galerkin methods for second-order elliptic problems, Math. Comp. 78 (2009), no. 265, 1 – 24. · Zbl 1198.65194
[14] Bernardo Cockburn, Weifeng Qiu, and Ke Shi, Conditions for superconvergence of HDG methods for second-order elliptic problems, Math. Comp. 81 (2012), no. 279, 1327 – 1353. · Zbl 1251.65158
[15] Bernardo Cockburn, Francisco-Javier Sayas, and Manuel Solano, Coupling at a distance HDG and BEM, SIAM J. Sci. Comput. 34 (2012), no. 1, A28 – A47. · Zbl 1241.65105
[16] Bernardo Cockburn and Manuel Solano, Solving Dirichlet boundary-value problems on curved domains by extensions from subdomains, SIAM J. Sci. Comput. 34 (2012), no. 1, A497 – A519. · Zbl 1238.65111
[17] B. Cockburn and M. Solano, Solving convection-diffusion problems on curved domains by extensions from subdomains, submitted. · Zbl 1304.65246
[18] Grégory Guyomarc’h, Chang-Ock Lee, and Kiwan Jeon, A discontinuous Galerkin method for elliptic interface problems with application to electroporation, Comm. Numer. Methods Engrg. 25 (2009), no. 10, 991 – 1008. · Zbl 1175.65136
[19] Anita Hansbo and Peter Hansbo, An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 47-48, 5537 – 5552. · Zbl 1035.65125
[20] R. Hiptmair, J. Li, and J. Zou, Convergence analysis of finite element methods for \?(\?\?\?;\Omega )-elliptic interface problems, J. Numer. Math. 18 (2010), no. 3, 187 – 218. · Zbl 1203.65227
[21] T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg. 194 (2005), no. 39-41, 4135 – 4195. · Zbl 1151.74419
[22] Randall J. LeVeque and Zhi Lin Li, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal. 31 (1994), no. 4, 1019 – 1044. · Zbl 0811.65083
[23] Randall J. LeVeque and Zhilin Li, Immersed interface methods for Stokes flow with elastic boundaries or surface tension, SIAM J. Sci. Comput. 18 (1997), no. 3, 709 – 735. · Zbl 0879.76061
[24] Adrian J. Lew and Matteo Negri, Optimal convergence of a discontinuous-Galerkin-based immersed boundary method, ESAIM Math. Model. Numer. Anal. 45 (2011), no. 4, 651 – 674. · Zbl 1269.65108
[25] Jingzhi Li, Jens Markus Melenk, Barbara Wohlmuth, and Jun Zou, Optimal a priori estimates for higher order finite elements for elliptic interface problems, Appl. Numer. Math. 60 (2010), no. 1-2, 19 – 37. · Zbl 1208.65168
[26] Y. Liu and Y. Mori, Properties of discrete delta functions and local convergence of the immersed boundary method. Submitted. · Zbl 1268.65143
[27] Yoichiro Mori, Convergence proof of the velocity field for a Stokes flow immersed boundary method, Comm. Pure Appl. Math. 61 (2008), no. 9, 1213 – 1263. · Zbl 1171.76042
[28] Charles S. Peskin, Flow patterns around heart valves, Proceedings of the Third International Conference on Numerical Methods in Fluid Mechanics (Univ. Paris VI and XI, Paris, 1972) Springer, Berlin, 1973, pp. 214 – 221. Lecture Notes in Phys., Vol. 19. · Zbl 0244.92002
[29] Theodore J. Rivlin, An introduction to the approximation of functions, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1969.
[30] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501
[31] Elias M. Stein and Rami Shakarchi, Real analysis, Princeton Lectures in Analysis, vol. 3, Princeton University Press, Princeton, NJ, 2005. Measure theory, integration, and Hilbert spaces. · Zbl 1081.28001
[32] Johan Waldén, On the approximation of singular source terms in differential equations, Numer. Methods Partial Differential Equations 15 (1999), no. 4, 503 – 520. , https://doi.org/10.1002/(SICI)1098-2426(199907)15:43.0.CO;2-Q · Zbl 0938.65112
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